There is no proof specifically this is really annoying about most of the Babylonian Texts, they never go into method very much.

However from the numbers they got (30, 1;24,51,10, and 30*1;24,51,10=42;25,35) it looks less like they used a proof as they did use an iterative approximation method as follows:

let a be some number

let a1 be an approximation of sqrt(a) such that a1 > sqrt(a)

then B1 = a/a1 is also an approximation of sqrt(a) but deviates in the other direction s B1 < sqrt(a).

we have now bracketed sqrt(a)

so a new better bracketing can now be made by

a2 = (a1 +B1)/2 and B2 = a/a2

and then

a3 = (a2 + B2)/2, B3 = a/a3

etc. the answer for sqrt(2) of 1;24,51,10 happens to equal a3 in this case.

note that the above is only speculation, but the method DOES produce the Babylonian numbers so and it uses only arithmetic that we know they had, so it seems pretty likely

(All this information and more can be found in Mathematical Cuneiform Texts by Otto Neugebauer)